Higher only · equation of a circle

Equation of a circle.
It is just
Pythagoras.

x² + y² = r²

a circle, centre the origin —

x² + y² = r²

every point is r from the centre

r is the radius

step 1 of 4

Pythagoras on a radius

1. Join any point on the circle to the centre.

2. 3 across, 4 up, radius 5: 3² + 4² = 5².

✓ x² + y² = 25

step 2 of 4

Read off the radius

1. The number on the right is r², not r.

2. Square root it to get the radius.

✓ Leave √10 as a surd

x² + y² = 25

r² = 25, so r = 5

x² + y² = 10

r = √10 not 10, not 5

step 3 of 4

Inside, on or outside?

1. (4, 3): 16 + 9 = 25. On the circle.

2. (2, 4): 4 + 16 = 20. Less than 25: inside.

3. (5, 2): 25 + 4 = 29. More than 25: outside.

✓ Substitute, then compare with r²

step 4 of 4

Tangent at (3, 4)

1. Radius gradient: rise ÷ run = 43.

2. The tangent is perpendicular: −34.

3. Through (3, 4): 4 = −94 + c, so c = 254.

✓ y = −34x + 254

circle, centre the origin —

x² + y² = r²

Pythagoras on the radius

Radius

square root the right-hand side

Tangent

perpendicular to the radius

Perpendicular gradient:
flip the fraction, change the sign.

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